Order-reversing Involution Research Articles

On the structure of the lattice of noncrossing partitions

We show that the lattice of noncrossing (set) partitions is self-dual and that it admits a symmetric chain decomposition. The self-duality is proved via an order-reversing involution. Two proofs are given of the existence of the symmetric chain decomposition, one recursive and one constructive. Several identities involving Catalan numbers emerge from the construction of the symmetric chain decomposition.

L-fuzzy topological groups

In this paper, we propose the concept of L-fuzzy topological groups and discuss their basic properties. Here L is a completely distributive lattice with order-reversing involution.

Fuzzy addition in the lowen fuzzy real line

For L a complete chain with order-reversing involution α→ α′, Rodabaugh [7] defined a fuzzy addition ⊛ in the Hutton fuzzy real line R( L) and studied some of its properties. Subsequently, Lowen [3] made a considerably simpler description of ⊛ and gave a direct proof of the continuity of ⊛. In this paper, we introduce the fuzzy addition ⊛ in the Lowen fuzzy real line μ <( R) [6]. We give several equivalent descriptions of ⊛ which may be used for investigating some properties of ⊛. Moreover, we prove the closedness of ⊛ in the four important subspaces A <( R), F <( R), μ < Z ( R), μ < Q ( R) [4] of μ <( R). This implies, in particular, that the sum of two fuzzy rational numbers is fuzzy rational.

A theory of fuzzy uniformities with applications to the fuzzy real lines

For each completely distributive lattice L with order-reversing involution, the fuzzy real line R (L) is uniformizable by a uniformity which both generates the canonical (fuzzy) topology and induces a pseudometric generating the canonical topology. If L is also a chain, the usual addition and multiplication defined on R  R ({0, 1}) extend jointly (fuzzy) continuously to ⊕ and ⊙ on R (L). Three fundamental questions in fuzzy sets until now are: Question A. If L 1≅ L 2, is R (L 1) uniformly isomorphic to R (L 2) in some sense? Question B. For each chain L, is ⊕ (jointly) uniformly continuous in a sense which guarantees its (joint) continuity on R (L)? Question C. Is R (L) a complete pseudometric space in some sense? We construct categories QU and U using the [quasi-] uniformities of B. Hutton which enable us to answer these questions in the affirmative. These results enhance the canonical standing of the fuzzy real lines and so give additional justification for answering in the affirmative: Question D. Does fuzzy topology have deep, specific, canonical examples?

L-fuzzy normal spaces and Tietze extension theorem

In this paper we prove a characterization theorem for normal L-fuzzy topological spaces ( L is an infinitely distributive lattice with an order-reversing involution). In the particular case L = {0, 1} this theorem reduces to a known result of Katětov ( Fund. Math. 38 (1951), 85–91) and Tong ( Duke Math. J. 19 (1952), 289–292). As an important application we obtain a fuzzy version of Tietze extension theorem. This yields an affirmative answer to a question raised in a series of papers by Rodabaugh ( Fuzzy Sets and Systems 11 (1983), 163–183).

Separation axioms and the fuzzy real lines

We show that for each lattice L which is completely distributive and equipped with an order-reversing involution, the canonical topology of the fuzzy real line ?(L) is uniformizable in the sense of Hutton; furthermore, this uniformity induces a pseudometric which induces the canonical topology. Consequently, ?(L) satisfies the higher order separation axioms of Hutton and Sarkar. More generally, we answer most of the open questions concerning ?(L) and its subspaces with respect to each of the separation axiom schemes of Hutton and Sarkar, answer some of these same questions for the stratified fuzzy real line ?c(L) and its subspaces, and analyze ?c(L) and its subspaces with respect to the separation axioms of Rodabaugh.